ISC 5935 Spring 2015 Study plan for Isaac Lyngaas Supervisor: John Burkardt 08 January 2015 -------------------------------------------------------------------------------- Isaac Lyngaas proposes the following special topic activity for his participation in the course. In a previous class, he was able to implement radial basis functions in 1D and 2D, for use in representing solutions of differential equations. One drawback to this approach, however, is that the corresponding system matrix for the coefficients is dense, meaning that the method becomes increasingly impractical for large numbers of data points or higher dimensions. He has seen papers describing a 'finite difference' radial basis function approach (Shankar, Flyer, Bollig), which modifies the radial basis functions so that the evaluation at any point only involves a small number of functions associated with nearby basis points. This makes the system matrix sparse, and frees the method to solve large or high dimensional problems. Isaac proposes to implement this new method; to test it on some standard benchmark problems (Franke, Renka); to write a report of his findings. If there is time, he would like to create a parallel version that can be run on our local HPC system. There is a possibility that, during this project, Isaac may wish to change the direction of his research; this is acceptable as long as we discuss it beforehand and can agree on a corresponding modification of this existing agreement. -------------------------------------------------------------------------------- Varun Shankar, Grady Wright, Robert Kirby, Aaron Fogelson, A Radial Basis Function (RBF)-Finite Difference (FD) method for Diffusion and Reaction-Diffusion Equations on Surfaces, arXiv:1404.0812v1 [math.NA] 3 Apr 2014 Natasha Flyer, Grady Wright, Bengt Fornberg, Radial Basis Function-generated Finite Differences: A Mesh-free Method for Computational Geosciences, Geoscience, Accepted. Evan Bollig, Natasha Flyer, Gordon Erlebacher, Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs, Journal of Computational Physics, Volume 231, Issue 21, pages 7133-7151, 20 August 2012. Richard Franke, A Critical Comparison of Some Methods for Interpolation of Scattered Data, Naval Postgraduate School Technical Report, NPS-53-79-003, 1979. Richard Franke, Scattered Data Interpolation: Tests of Some Methods, Mathematics of Computation, Volume 38, Number 157, January 1982, pages 181-200. Robert Renka, Ron Brown, Algorithm 792: Accuracy Tests of ACM Algorithms for Interpolation of Scattered Data in the Plane, ACM Transactions on Mathematical Software, Volume 25, Number 1, March 1999, pages 78-94.